The nucleus of the atom was discovered by Ernest Rutherford in 1911. He conducted the famous gold foil experiment, where alpha particles were directed at a thin sheet of gold. Observing that some particles were deflected at large angles, Rutherford concluded that the atom has a tiny, dense, positively charged nucleus where most of its mass is concentrated, surrounded by mostly empty space where electrons orbit. This discovery replaced the earlier "plum pudding model" of the atom.
It is important to note that Robert Brown, a Scottish botanist, discovered the nucleus of the cell in 1833, which is distinct from the atomic nucleus discovered by Rutherford.